Question

Let A = R−{0}, the set of all nonzero real numbers, and consider the following relations on A × A. Decide in each case whether R is an equivalence relation, justifying your answers. In the case that R is an equivalence relation, find the equivalence classes. (a) (a, b)R(c, d) if and only if ad = bc

Answer 1

Answer:

Step-by-step explanation:

Let A = R−{0}, the set of all nonzero real numbers, and consider the following relations on A × A.

Given that (a,b) R (c,d) if $ad=bc$

Or (a,b) R (c,d) if determinant

$\left[\begin{array}{ccc}a&b\\c&d\end{array}\right] =0$

a) Reflexive:

We have (a,b) R (a,b) because ab-ab =0 Hence reflexive

b) Symmetric

(a,b) R (c,d) gives ad-bc =0

Or da-cb =0 or cb-da =0 Hence (c,d) R(a,b). Hence symmetric

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